Marginal Effects of a Fractional Response Model

[Hakan Gunduz]

Hi all,

I have a Fractional Response Model where my dependent variable is bounded between 0 and 1 - with a lot of zeros.

Some of my independent variables lie between 0 and 1 as well, but not all of them.

Should I go for

margins, dydx(*)

or

margins, dyex(*)

?

Here is the output as well:

 fracreg logit y x1 x2 x3 x4 if datayearfiscal==2008

Iteration 0:   log pseudolikelihood = -197.08665  
Iteration 1:   log pseudolikelihood = -182.02928  
Iteration 2:   log pseudolikelihood = -181.99607  
Iteration 3:   log pseudolikelihood = -181.99607  

Fractional logistic regression                  Number of obs     =        279
                                                Wald chi2(4)      =      87.36
                                                Prob > chi2       =     0.0000
Log pseudolikelihood = -181.99607               Pseudo R2         =     0.0589

------------------------------------------------------------------------------
             |               Robust
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          x1 |   1.254019   .3392668     3.70   0.000     .5890682     1.91897
          x2 |  -2.369368   .9091443    -2.61   0.009    -4.151258   -.5874784
          x3 |   .1100148   .1341461     0.82   0.412    -.1529068    .3729364
          x4 |  -2.741405   .4381923    -6.26   0.000    -3.600246   -1.882564
       _cons |   .2069838   .2960355     0.70   0.484    -.3732351    .7872028
------------------------------------------------------------------------------

x1, x2, and x3 lie between 0 and 1 while x4 does not have bounds.

 margins, dydx(*)

Average marginal effects                        Number of obs     =        279
Model VCE    : Robust

Expression   : Conditional mean of y, predict()
dy/dx w.r.t. : x1 x2 x3 x4

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          x1 |   .2888261   .0765901     3.77   0.000     .1387123    .4389399
          x2 |  -.5457139      .2072    -2.63   0.008    -.9518185   -.1396092
          x3 |   .0253387   .0308573     0.82   0.412    -.0351405    .0858178
          x4 |  -.6314015   .0943868    -6.69   0.000    -.8163962   -.4464069
------------------------------------------------------------------------------

 margins, dyex(*)

Average marginal effects                        Number of obs     =        279
Model VCE    : Robust

Expression   : Conditional mean of y, predict()
dy/ex w.r.t. : x1 x2 x3 x4

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/ex   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          x1 |   .1726102   .0459446     3.76   0.000     .0825605    .2626599
          x2 |   -.061035   .0229026    -2.66   0.008    -.1059233   -.0161466
          x3 |   .0125208   .0152707     0.82   0.412    -.0174091    .0424508
          x4 |  -.1666824   .0238142    -7.00   0.000    -.2133574   -.1200074
------------------------------------------------------------------------------

Is the following correct for the dyex?
1% increase in x1 increases the value of y by 0.173

Does that make sense to talk about 1% change in x1 while it is already in percentage as well. Would it be better to go for dydx, instead?

Thanks a lot.

[Maarten Buis]

You need to make the distinction between a percentage change (dyex) and a percentage point change (dydx) very clear. If you start with 10% and increase that by 1 percent, then you end up with 10%*1.01=10.1%. If you start with 10% and increase that by 1 percentage point, then you end up with 10% + 1% = 11%.

In principle using percentage changes on a variable that is itself a percentage is perfectly fine. In practice it will be a source of endless confusion for you and your audience. I don't write my articles, papers, presentations in one go, so it is easy to forget the minor details somewhere along the line. This is a case where you cannot afford to slip up, as any slip will cause the audience to become hopelessly confused and may well turn them against your paper. Similarly, while presenting this is something where you can easily slip up, which is something you cannot afford to do. Even if I get this perfectly right, there will always be multiple persons in the audience that mixes them up (and I can't blame them). So just for my own peace of mind I would always choose dydx (i.e. percentage points), so I never have to worry about these mistakes.